**Gaussian Elimination (Row Reduction)**

Gaussian elimination is an algorithm for solving systems of linear equation. It is avaible when the number of equations equals or more than the number of unknowns. For example;

1x + 3y + 1z = 9

x + y – z = 1

y +2z = 35

There are 3 unknowns and 3 equations so, we can apply gaussian elimination to reach the set of solution. Let’s see..

First we have to write the equation system as a matrix. Then continue row operations until obtain a matrix form like the given below.

This form is also called **row echelon form**. Also it is called **reduced row echelon form**, if there are zeros instead of * (stars). After obtain the row echelon form, we can reach the set of solution.

**Gauss-Jordan Elimination**

The only difference between Gaussian Eliminaton and Gauss-Jordan Elimination is continue to row reduction operations until obtain **reduced row echelon form** instead of *row echelon form* so, we reach the set of solution directly.

**Inverse of a Matrix**

There are different ways to find the inverse of a matrix. The first way is **finding inverse of a matrix with elementary row operations**. Write the given matrix with an identity matrix as an augmented matrix like the given below;

The given matrix is on left side and the identity matrix on right side. We are continue to elementary row operations *until obtain an identity matrix on the left side*. When we obtain an identity matrix on the left side, we obtain inverse of the given matrix on the right side. Thats all.

Also we can find inverse of a given matrix with the formula given below;

As seen at the formula, inverse of a matrix A is equal to multiplication of **adjacency matrix** of A with 1 over **determinant** of matrix A. Also adjacency matrix is equals to **transpose** of **coefficient matrix** of A.